# Group in which every endomorphism is trivial or an automorphism

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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The version of this for finite groups is at:finite group in which every endomorphism is trivial or an automorphism

## Definition

A **group in which every endomorphism is trivial or an automorphism** is a (typically, nontrivial) group for which every endomorphism is either the trivial map (sending all group elements to the identity element) or is an automorphism.

Whether the trivial group is included or not is a matter of convention. We sometimes exclude it when comparing with other simple group-type properties.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite simple group | finite, nontrivial, and no proper nontrivial normal subgroup | finite simple implies every endomorphism is trivial or an automorphism | | | |

finite quasisimple group | finite, perfect, and inner automorphism group is simple | finite quasisimple implies every endomorphism is trivial or an automorphism | | | |

simple co-Hopfian group | simple and not isomorphic to any proper subgroup | simple co-Hopfian implies every endomorphism is trivial or an automorphism | | |

### Weaker properties conditional to nontriviality

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

directly indecomposable group | nontrivial and not expressible as a direct product of nontrivial groups | Splitting-simple group|FULL LIST, MORE INFO | ||

splitting-simple group | nontrivial and has no proper nontrivial complemented normal subgroup | every endomorphism is trivial or an automorphism implies splitting-simple | | |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group in which every endomorphism is trivial or injective | every endomorphism is trivial or injective; equivalently, it has no proper nontrivial endomorphism kernel | | | ||

Hopfian group | every surjective endomorphism is an automorphism | Group in which every endomorphism is trivial or injective|FULL LIST, MORE INFO | ||

co-Hopfian group | every injective endomorphism is an automorphism | | |